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Why is the Poisson equation often written as: $-\Delta u=f$ ? Similarily, the Sturm equation is often written as: $-(pu')'+qu=f$. What purpose does the minus sign serve? Thanks

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  • $\begingroup$ the minus sign is to make the operator selfadjoint and so all the eigenvalues are always real and positive as a counter example imagine $ \frac{d^{2}f}{dx}= \lambda_{n}f(x) $ with conditions $ y(0)=y(1)=0$ are the eigenvalues real $\endgroup$ – Jose Garcia Feb 3 '14 at 12:59
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Because it is convenient to do some theoretical analysis especially about weak form. It will be cancel by using integral by parts in most cases.

For example, in the Poisson equation, assume $u\in H^1(\Omega)$, where $\Omega$ is the domain of equation and $H$ donate classical Sobolev space. We can obtain the weak form of the equation by the following deduction:

\begin{equation} (-\Delta u,v)\stackrel{\mathrm{Green's\ identity}}{=}(\nabla u,\nabla v)=(f,v) \quad \forall v\in H^1_0(\Omega) \end{equation} Where $(u,v)=\int_\Omega u v d\sigma$ is the classical $L^2$ inner product, and $v$ is the test function. In this deduction, the minus sign have been cancel by using Green's identity. Thus, the weak form of this equation have a pretty statement.

In the Sturm equation, the minus sign have a same function. We have following weak form about it: \begin{equation} (−(pu')',v)+(qu,v)\stackrel{\mathrm{Integral\ by\ parts}}{=}(pu',v')+(qu,v)=(f,v)\quad \forall v\in H_0^1(\Omega) \end{equation} So the minus sign in the equation is used to simplify the process of analysis. You can find more books about Sobolev space, ODE and PDE and they will give you more details.

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